Question 2 (b)
Optimal replacement time of a machine
9 marks
- Time to write
- About 18 minutes
Question
Question 2 (b) · 9 marksA machine costs ₹12,480. Up to a time t from the date of purchase, the total cost of operation is given by the function: 70 t² + 0.3 t. If the machine is sold as scrap after t years, its resale value is described by the function 7800 - 60 t². Find the optimal time to replace the machine.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- replaceCash flows year by year in one table, then discount, then a one-line decision.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Total and average cost functions 3 marks 5 min
- 2 First-order condition 3 marks 5 min
- 3 Second-order condition 2 marks 4 min
- 4 Replacement time 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Average cost per year. If the machine is replaced after t years, the total cost borne is the capital lost (cost less resale value) plus the operating cost up to t:
Total cost = 12,480 − (7,800 − 60t²) + 70t² + 0.3t = 4,680 + 130t² + 0.3t.
Average annual cost A(t) = (4,680 + 130t² + 0.3t)/t = 4,680/t + 130t + 0.3 (average cost = total cost ÷ time).
dA/dt = −4,680/t² + 130 = 0 ⟹ t² = 36 ⟹ t = 6 years. d²A/dt² = 9,360/t³ > 0 for t > 0, so A is a minimum at t = 6 (derivatives by the power rule).
| At t = 6 years | ₹ |
|---|---|
| Capital lost: 12,480 − (7,800 − 60 × 36) = 12,480 − 5,640 | 6,840.00 |
| Operating cost: 70 × 36 + 0.3 × 6 | 2,521.80 |
| Total cost | 9,361.80 |
Average cost = 9,361.80/6 = ₹1,560.30 a year (against ₹1,586.30 at 5 years and ₹1,578.87 at 7 years).
The optimal time to replace the machine is after 6 years.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Total and average cost functions | 3 |
| 2 | First-order condition | 3 |
| 3 | Second-order condition | 2 |
| 4 | Replacement time | 1 |
| Total | 9 |
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